Building Strong Shapes with Triangles

A Guided Participatory Lesson for Children

WARNING FOR PARENTS: Parts of this lesson involve the use of scissors and sharp tools for making small holes in cardboard or posterboard. Please make sure that your child is either properly instructed not to use these tools, or is properly supervised as appropriate for their age, in order to maximize their safety. As a parent it is your responsibility to ensure that you control exposure of your child to instructions advising the use of potentially dangerous tools, and that you provide any age-appropriate supervision or instructions. It is our desire that children benefit from this material in a safe and productive way.

MESSAGE FOR CHILDREN: Some parts of these lessons involve the use of scissors and other sharp tools for making small holes in cardboard or posterboard. Do not use scissors, or any sharp tools to make the holes by yourself, unless you have permission from a parent! It is better and smarter to ask for help than to use something dangerous by yourself! Ask a parent to look at this web page with you to decide how you can safely use it. We want you to be safe while you learn!

For Parents: This is a guided participatory lesson for young children about simple shapes and how strength may be added to structures by building with triangles. The lesson starts with two-dimensional shapes, and then proceeds to three-dimensional shapes. Parents, if working with your child, you may wish to break this material into several sessions, or exclude the more advanced material you feel may be beyond your child's grade level. However, you are encouraged to let your child seek their own level, and you may be surprised to discover how much of this material comes naturally to them! This material will be refined and developed into a lesson plan at a future date.

If used as the basis for a science project, the student can use this material to learn about the subject, and as source material to include in the final presentation. However, it will be the job of the student to incorporate these lessons and concepts into the format of a science project presentation as specified by the teacher.

Two-dimensional Shapes

Two-dimensional objects are flat. That is, they can be simply drawn on a piece of paper, or lie flat on a table top. Our world is filled with many two-dimensional shapes including circles, triangles, squares, and many others.

Many shapes, like this one, are too complicated to have a simple name.

When people build things, they use many different shapes, because every shape has special characteristics that are best suited for a particular purpose. For example, a wheel on a car, and a Ferris wheel, both use circles, because a circle turns nicely.

The metal or wood beams that hold up most houses and buildings form rectangles and squares. Here are two buildings where the walls have not yet been finished. You can easily see all of the squares and rectangles inside. A few of them have been highlighted with yellow.

Squares and rectangles always have four sides.

Triangles may be found in many bridges, and help to make them strong, as we will shortly see. Here are a couple of examples of bridges that have many triangles.

The Disney Epcot Center dome is made entirely of triangles, which keep it very strong.

Polygons

Shapes that have sides that are all straight lines are called Polygons. You can draw polygons on a piece of paper. You can also build them out of wooden sticks, pipe cleaners, or other straight objects.

Polygons may have three, four, five, six, or more sides. Many of the simple polygons have been given names. For example a polygon with three sides is called a triangle. A polygon with four sides is called a square. A polygon with five sides is called a pentagon, because pent means five. Hex means six, so a hexagon has six sides. A polygon can be made of lines and be hollow, like these shapes which are simply drawn on a piece of paper.

You can also make polygons using Roger's Connection. Try building these shapes now. You can use any colors that you like.

What is an Angle?

Let's connect two lines together at one end like one of the examples in the picture below. In these examples, the two blue lines in each example are connected at the black dots. Let's leave the other ends of the blue lines free to move. You can see that the lines can change their positions with respect to one another, even though they stay connected at the black dot. If one of the lines stays in the same place, the other line can spin in a circle, just like the hands of a clock. The word Angle describes how far apart the two lines are from each other, in terms of how much of a full circle one line is separated from the other. In these three examples we show a small angle, a bigger angle, and a very big angle. There is a way of assigning a number to each angle, but we don't need to understand that idea right now.

Regular Polygons

The polygons on the left below don't look as neat as the ones we made just before. When all of the sides of a polygon are the same length, and the angles between each side are the same, then a polygon is called a Regular Polygon. We just learned about the word angle. Imagine that the steel balls below are like the black dots in the picture above when we discussed angles. Now you can see that each of the shapes below in the left picture contains a mixture of different angles. When each polygon is adjusted so that all of the angles within it are the same, then it is called a regular polygon. The polygons in the picture on the left are polygons, but they are NOTregular polygons! The polygons in the picture on the right ARE regular polygons.

These are NOT Regular Polygons

These are Regular Polygons

Note: Earlier we said that a polygon with four sides was called a square, but now we can tell you that only a regular polygon with four sides is called a square. The blue four sided polygon in the picture on the left above is not a square. It is called a parallelogram. It has four sides of equal length, but all the angles are not the same. Two of the angles are small, and two are large. Parallel means going in the same direction. In the parallelogram, each pair of opposite sides is going in the same direction.

Polygons can also be filled in like these shapes cut from paper. Can you name each shape?

A polygon can never have curves in it, so this shape is NOT a polygon.

When you built the polygons earlier with Roger's Connection, did you notice anything special about the triangle? Did it seem like it was stronger than the other shapes? While all of the other polygons can be bent into many different forms that are NOT regular polygons (with many different angles in each polygon), the triangle always keeps the same shape. It is the strongest polygon. Why is that? The reason is because in all of the other polygons, all of the angles can change. There is nothing to stop them. However in the triangle, the angles can not change once the triangle is built.. The angles are fixed. This is because a triangle has three sides and three angles, and each angle is fixed by the side opposite to it. If you look at the following picture, you can see that there is only one angle where the two free sides can connect to the third side, and that once connected, all of the angles are fixed. Try this with your Roger's Connection triangle now. Then make the different polygons with four, five, and six sides, and see how the triangle is the only one that can't be adjusted into a different shape once it is made.

Now that you see how strong triangles are, and why they are special among polygons, you can start to understand why people build with triangle when great strength is needed, just like in the examples of the bridges and the Epcot dome shown in the earlier pictures. If those bridges had been made with only squares, they would not have been very strong at all.

Strengthening Squares

We have seen how the triangle is strong, and how the other polygons can not hold their shape as easily. Here is a short movie showing how the square is not able to resist being changed.

Can a square be made stronger by adding triangles? The answer is yes. If you start with a square, you can add a diagonal between opposite corners to make it very strong. The word diagonal means something that goes between two opposite corners. By adding a diagonal, you actually make two triangles inside of the square. Since each triangle is strong, the new reinforced square is stronger as well. The word reinforced means to make stronger.

If you take your Roger's Connection square and try to make it stronger by adding a diagonal, you will find that you can only do it if you bend the square as shown below.

Here is a movie showing the same thing:

Adding the diagonal (the yellow magnetic rod in this picture) makes the shape stronger, but it is no longer a square. Why is that? The reason is because currently, the Roger's Connection magnetic rods only come in one length. If you wanted to keep the square shape and strengthen it with a Roger's Connection diagonal, you would need a longer yellow piece which we do not currently have!

The shape you did make in the last picture above is called a parallelogram. Parallel means going in the same direction. In the parallelogram, each pair of opposite sides is going in the same direction. A parallelogram is a four-sided polygon where each of the four sides is the same length.

However, there is a way you can show how to make a stronger square, using cardboard and paper fasteners instead of Roger's Connection. First, get some cardboard or posterboard, and four paper fasteners. Cut four pieces five and a half inches long and one inch wide. Cut another piece ten inches long and one inch wide. Carefully make holes in the ends of each short piece, centered, and one half inch from the ends. Make just one hole in one end of the long piece. Make the holes just big enough to allow the paper fasteners to fit. Ask a parent to help you cut the cardboard or posterboard, and make the holes, so that this can be done safely. Do not use scissors or any sharp tools to make the holes by yourself unless you have permission from a parent! It is better and smarter to ask for help than to use something dangerous by yourself! Here is what these pieces should look like when you are finished. This picture also shows the paper fasteners you will need.

Next, use the paper fasteners and four of the five pieces to make a square as shown.

Hold two of the opposite (diagonal) corners and move them back and forth to see how flexible and unsupported this square is, as shown in the movie below.

Just in case you can't see the movie, here are some pictures of some of the different ways the square can be distorted. The word distorted means that something has been changed from how it was in it's original natural state.

Next, remove one of the fasteners and add the fifth long piece as shown below, reconnecting the corner. Note that the long piece doesn't have a second hole at the other end yet. Also note that the long piece is covering up one of the fasteners at the bottom right of the picture. At this point, you can still adjust the square to many different shapes (many different parallelograms). Adjust the pieces so that they look like the picture below.

Now we want to make the square strong by using triangles. Take a pencil or pen, and make a small dot on the bottom right of the long piece, just above the hidden fastener underneath. In the picture below, the dot is blue.

Next, again with the help of a parent if you don't have permission to use a sharp tool by yourself, make a hole like the others at the dot you just made in the long piece. Then, remove the fastener underneath, and put that corner back together, now also going through the hole in the long piece that you just made. When you are done it should look like this:

Finally, if you like, you can cut off the extra length of the long piece as shown below to make it neater:

Now try to adjust the square back into a parellelogram. Right away, you can see that the square is much stronger, and you can no longer move it as before (unless you bend the paper). This idea of making weak squares much stronger is used in the real world all of the time. Look again at the picture of the second bridge. You will see many shapes that are similar to squares and rectangles that have been made much stronger using exactly the same technique that we just used above. (A technique is a way of doing something.) In this technique, we added a diagonal that connects between two opposite corners of a square. This method works just as well for rectangles. This is one example of special knowledge that engineers and designers use to build things in the real world, like buildings, bridges, and airplanes, that are strong and safe to protect the people that use them. Remember all of the squares and rectangles in the earlier pictures of buildings being constructed? Here is a picture of a similar building, in which triangles have been added to make the building stronger, exactly as we have made squares stronger by adding a diagonal.

By building with Roger's Connection, you will naturally learn this principle, and discover that many of your designs can be made stronger by building with triangles. You will also discover many other important ideas about shapes and how they are made strong so that they can be used in practical ways in the real world.

Strengthening Other Polygons with Triangles

We can make a square stronger in another way. Here we start with a four-sided polygon, a parallelogram, and show how it can be distorted. Like before, it doesn't start out very strong. Next, we adjust the parallelogram to make a square. And finally, we add the four yellow magnetic rods to make a little pyramid. This has made the square stronger by using triangles in another way. You will find that there are many ways to use triangles to make shapes stronger. In this example, we have also built our first three-dimensional shape. Two-dimensional shapes lie flat on a table or on a piece of paper, while three-dimensional shapes rise up above the table as in the pyramid below.

Can other polygons can be made stronger using triangles? Yes. Using Roger's Connection, we can make a make a five-sided polygon stronger in a similar way to what we just did with a four-sided polygon. First we make a five-sided polygon and show how easily it can be changed, and is not very strong. Next we adjust it to be regular polygon, a pentagon, as shown in the middle picture below. And finally we add five more magnetic rods shown in yellow below. By creating these five triangles, we have made the pentagon much stronger. The final three-dimensional shape we just made is another kind of pyramid. The fancy name for this shape is a pentagonal pyramid. Pent means five, and the pentagon has five sides.

Now let's try this with a six-sided polygon. First we show how the six-sided polygon is not very strong by itself. Then we adjust it to make it a regular polygon, a hexagon. And finally we add six magnetic rods which makes the hexagon much stronger by forming six triangles.

Hexagons

Did you notice something different about this shape? When we added the six magnetic rods the shape stayed flat on the table, and remained a two-dimensional shape instead of becoming a three-dimensional shape. The hexagon is a special shape that has this unique property. Hexagons can also contain circles very neatly, by placing one in each triangle.

Hexagons also fit together very nicely when put side-by-side, into an arrangement called a hexagonal grid.

And this hexagonal grid can also contain circles in another very compact way.

Because of all of these special characteristics of hexagons, nature uses hexagons in many different way. In the photos below, you can see two examples. On the left, you can see the many hexagons in a honeycomb where bees store honey and take care of their young. In the picture on the right, you can see part of the eye of a fly as seen through a microscope. Each little bump is sensitive to light so the fly can see.

Three-Dimensional Shapes

So far, we have been talking about shapes that are flat and can be drawn on a piece of paper, like the triangle, square, pentagon, and hexagon. But many things in the real world are not flat at all. How can we take the basic flat polygon shapes, and make three-dimensional shapes that aren't flat? This can be done very simply, or it can be very complicated. Let's take a look at some simple but interesting examples.

Build a Tetrahedron

This is the simplest three-dimensional shape, called a tetrahedron. You have actually already built it in an earlier part of this lesson. Let's build it again, and discover several different ways to make it. Tetra means four, and the tetrahedron has four sides which are all triangles, so the tetrahedron is very strong. Click here to learn how to make the tetrahedron in several different ways. When you want to return to this web page to continue, click on the Back button at the top of your web browser. Have you made the tetrahedron? You will notice that when you pick up this shape and turn it in any direction, all of the triangles look the same, and you can't really tell which one you built first anymore (unless you used several colors). This shape is called a tetrahedron because tetra means four. The tetrahedron has four sides that are all triangles. Remember the word polygon? That referred to flat shapes. In the same way the word polyhedron, refers to a three-dimensional shapes - that is, a shape that is not flat, and has straight edges. With Roger's Connection, magnetic rods take the place of edges. Just as there are many kinds of polygons, there are also many kinds of polyhedra. The first and simplest polyhedron that you just made is called a tetrahedron. Because a tetrahedron is made entirely of triangles, it is very strong, and keeps its shape well. In the same way that triangles make strong flat shapes, they also make strong shapes in three-dimensions. The atoms inside a diamond are arranged as many connected tetrahedrons, and that is why a diamond is such a hard and strong material.

Build a Cube

Can we build a polyhedron with square sides instead of triangles? Let's try, and build a cube. A cube is like a three-dimensional square, or a box shape.

Let's start with a square.

Next, add four more pieces like this:

Finally, add a top square like this.

If you have been successful, you have made what is called a cube. The cube is made of six squares. You probably discovered that this was not easy to do! The cube we made for this photograph wanted to fall apart very badly, and I had to steady it many times to make this picture! Remember how unsteady the square was? That unstable character shows itself again in the cube. Six unsteady squares combine to make a very unsteady cube! It is very easy to distort.

Now remember again how strong the tetrahedron was. Now you can see that building with triangles, even in three dimensions, results in very strong designs, and that building with squares results in much weaker designs. Often when building with squares, designers and engineers try to make them stronger by adding diagonals to form triangles.

Build a Two-frequency Tetrahedron

Let's make another design using triangles that is also very strong, and a very attractive design too. Let's make a bigger tetrahedron. Here are step-by-step instructions. We will use different colors to make the steps easy to follow. First make this large triangle made of three smaller triangles.

Next, build up the red triangle into a tetrahedron.

Now, build up the blue triangle into a tetrahedron.

And now build up the green triangle into a tetrahedron.

Next, add three more pieces like these in yellow to build another triangle..

And finally, complete the yellow tetrahedron at the top.

Congratulations! This design is officially called a two-frequency tetrahedron. Two-frequency basically means that it is two levels high. If you had enough pieces, you could make a three-frequency or larger tetrahedron, which would also be very strong.

Build an Octahedron

Click here to learn how to make the octahedron in several different ways. When you want to return to this web page to continue, click on the Back button at the top of your web browser. Have you made the octahedron? As you rotate the octahedron in your hands, notice that it contains three squares and eight triangles. Can you find them?This shape is called an octahedron because octa means eight. The tetrahedron has eight sides that are all triangles.

Build a Truss

Let's create another design that uses triangles to make squares stronger. Let's make something called a truss. Here are the steps:

First make three connected squares as follows. You will find that they can be changed into a parallelogram as a group, and are not very strong.

Next add four blue magnetic rods as shown and connect them at the top.

Next add four yellow magnetic rods as shown and connect them at the top.

Next add four red magnetic rods as shown and connect them at the top.

Finally add two more purple magnetic rods as shown on the top. and you have finished building the truss!

Truss structures similar to the one above are used when designers need to build something that is long and strong and lightweight, like the trusses used in the space station design below.

Build an Icosahedron

Click here to learn how to make the icosahedron. When you want to return to this web page to continue, click on the Back button at the top of your web browser. Have you made the icosahedron? As you rotate the octahedron in your hands, notice that it contains twenty triangles. The beautiful icosahedron is more complicated and difficult to make than the others and will take some patience to build.

Conclusion

Using Roger's Connection, you can build many, many different shapes. If you remember to try to use triangles in your designs, then your designs will be strong. Without triangles, a design will be much weaker, or sometimes even impossible. One day you may be a designer or engineer and create things for other people to use that make use of some of the shapes we have been discussing. The lessons you learn today about these shapes and the use of triangles, may help you to create stronger and more successful designs in the future. Many other people have had to learn these lessons only from books. If you have followed these examples then you have a big advantage in having learned these lessons first-hand. This will help you remember these lessons much more easily!